Noname manuscript No. (will be inserted by the editor) A mathematical model for breath gas analysis of volatile organic compounds with special emphasis on acetone Julian King · Karl Unterkofler · Gerald Teschl · Susanne Teschl · Helin Koc · Hartmann Hinterhuber · Anton Amann Received: date / Accepted: date J. King Breath Research Institute of the Austrian Academy of Sciences Dammstr. 22, A-6850 Dornbirn, Austria E-mail: [email protected] K. Unterkofler Vorarlberg University of Applied Sciences and Breath Research Unit of the Austrian Academy of Sciences Hochschulstr. 1, A-6850 Dornbirn, Austria E-mail: karl.unterkofl[email protected] G. Teschl University of Vienna, Faculty of Mathematics Nordbergstr. 15, A-1090 Wien, Austria E-mail: [email protected] S. Teschl University of Applied Sciences Technikum Wien Hochst¨ adtplatz¨ 5, A-1200 Wien, Austria E-mail: [email protected] H. Koc Vorarlberg University of Applied Sciences Hochschulstr. 1, A-6850 Dornbirn, Austria H. Hinterhuber Innsbruck Medical University, Department of Psychiatry Anichstr. 35, A-6020 Innsbruck, Austria A. Amann (corresponding author) Innsbruck Medical University, Univ.-Clinic for Anesthesia and Breath Research Institute of the Austrian Academy of Sciences Anichstr. 35, A-6020 Innsbruck, Austria E-mail: [email protected], [email protected] 2 Abstract Recommended standardized procedures for determining exhaled lower res- piratory nitric oxide and nasal nitric oxide (NO) have been developed by task forces of the European Respiratory Society and the American Thoracic Society. These rec- ommendations have paved the way for the measurement of nitric oxide to become a diagnostic tool for specific clinical applications. It would be desirable to develop similar guidelines for the sampling of other trace gases in exhaled breath, especially volatile organic compounds (VOCs) which reflect ongoing metabolism. The concentrations of water-soluble, blood-borne substances in exhaled breath are influenced by: – breathing patterns affecting gas exchange in the conducting airways – the concentrations in the tracheo-bronchial lining fluid – the alveolar and systemic concentrations of the compound. The classical Farhi equation takes only the alveolar concentrations into account. Real- time measurements of acetone in end-tidal breath under an ergometer challenge show characteristics which cannot be explained within the Farhi setting. Here we develop a compartment model that reliably captures these profiles and is capable of relating breath to the systemic concentrations of acetone. By comparison with experimental data it is inferred that the major part of variability in breath acetone concentrations (e.g., in response to moderate exercise or altered breathing patterns) can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. Moreover, it is deduced that measured end-tidal breath concentrations of acetone determined during resting conditions and free breathing will be rather poor indicators for endogenous levels. Particularly, the current formulation includes the classical Farhi and the Scheid series inhomogeneity model as special limiting cases and thus is expected to have general relevance for other classes of volatile organic compounds as well. Our model is a first step towards new guidelines for breath gas analysis of acetone (and other water-soluble compounds) similar to those for nitric oxide. Keywords breath gas analysis · volatile organic compounds · acetone · modeling Mathematics Subject Classification (2000) 92C45 · 92C35 · 93C10 · 93B07 3 1 Introduction Measurement of blood-borne volatile organic compounds (VOCs) occurring in ex- haled breath as a result of normal metabolic activity or pathological disorders has emerged as a promising novel methodology for non-invasive medical diagnosis and therapeutic monitoring of disease, drug testing and tracking of physiological pro- cesses [2,3,1,61,47]. Apart from the obvious improvement in patient compliance and tolerability, major advantages of exhaled breath analysis compared to conven- tional test procedures, e.g., based on blood or urine probes, include de facto unlim- ited availability as well as rapid on-the-spot evaluation or even real-time analysis. Additionally, it has been pointed out that the pulmonary circulation receives the en- tire cardiac output and therefore the breath concentrations of such compounds might provide a more faithful estimate of pooled systemic concentrations than single small- volume blood samples, which will always be affected by local hemodynamics and blood-tissue interactions [54]. Despite this huge potential, the use of exhaled breath analysis within a clinical setting is still rather limited. This is mainly due to the fact that drawing reproducible breath samples remains an intricate task that has not fully been standardized yet. Moreover, inherent error sources introduced by the complex mechanisms driving pulmonary gas exchange are still poorly understood. The lack of standardization among the different sampling protocols proposed in the literature has led to the development of various sophisticated sampling systems, which selectively extract end-tidal air by discarding anatomical dead space volume [34,25,13]). Even though such setups present some progress, they are far from being perfect. In particular, these sampling systems can usually not account for the variability stemming from varying physiological states. In common practice it is tacitly assumed that end-tidal air will reflect the alveo- lar concentration CA which in turn is proportional to the concentration of the VOC in mixed venous blood Cv¯ , with the associated factor depending on the substance- specific blood:gas partition coefficient lb:air (describing the diffusion equilibrium be- tween capillaries and alveoli), alveolar ventilation V˙A (governing the transport of the compound through the respiratory tree) and cardiac output Q˙c (controlling the rate at which the VOC is delivered to the lungs): Cv¯ Cmeasured = CA = : (1) V˙A lb:air + Q˙c This is the familiar equation introduced by Farhi [19], describing steady state in- ert gas elimination from the lung viewed as a single alveolar compartment with a fixed overall ventilation-perfusion ratio V˙A=Q˙c close to one. Since the pioneering work of Farhi, both equalities in the above relation have been challenged. Firstly, for low blood soluble inert gases, characterized by lb:air ≤ 10, alveolar concentra- tions resulting from an actually constant Cv¯ can easily be seen to vary drastically in response to fluctuations in blood or respiratory flow. While this sensitivity has been exploited in MIGET (Multiple Inert Gas Elimination Technique, cf. [79,78]) to assess ventilation-perfusion inhomogeneity throughout the normal and diseased lung, it is 4 clearly problematic in standard breath sampling routines based on free breathing, as slightly changing measurement conditions (regarding, e.g., body posture, breathing patterns or stress) can have a large impact on the observed breath concentration [17]. This constitutes a typical example for an inherent error source as stated above, poten- tially leading to a high degree of intra- and consequently inter-individual variability among the measurement results [53]. It is hence important to investigate the influ- ence of medical parameters like cardiac output, pulse, breathing rate and breathing volume on VOC concentrations in exhaled breath. In contrast, while highly soluble VOCs (lb:air > 10) tend to be less affected by changes in ventilation and perfusion, measurement artifacts associated with this class of compounds result from the fact that – due to their often hydrophilic properties – a substantial interaction between the exhalate and the mucosa layers lining the conducting airways can be anticipated [6]. In other words, for these substances Cmeasured 6= CA, with the exact quantitative re- lationship being unknown. Examples of endogenous compounds that are released into the gas phase not only through the blood-alveolar interface, but also through the bronchial lining fluid are, e.g., acetone and ethanol [7,76]. The emphasis of this paper lies on developing a mechanistic specification of end-tidal breath dynamics of highly soluble, blood-borne trace gases during differ- ent physiological situations (e.g., rest, exercise, sleep and exposure scenarios) using a prototypic compound. Our main focus will be on the ketone body acetone (2 - propanone; CAS number 67 - 64 - 1; molar mass 58:08 g/mol), which is one of the most abundant VOCs found in human breath and has received wide attention in the literature. Being a natural metabolic intermediate of lipolysis [31], endogenous ace- tone has been considered as a biomarker for monitoring the ketotic state of diabetic and fasting individuals [73,57,60,66], estimating glucose levels [20] or assessing fat loss [38]. Nominal levels in breath and blood have been established in [80,65], and bioaccumulation has been studied in the framework of exposure studies and pharma- cokinetic modeling [84,37,49]. Despite this huge body of experimental evidence, the crucial link between breath and blood levels is still obscure. For instance, multiply- ing the proposed population mean of approximately 1 mg/l [65] in end-tidal breath by the partition coefficient lb:air = 340 [7] at body temperature appears to grossly underestimate observed (arterial) blood levels spreading around 1 mg/l [80,84,33]. Furthermore, continuous profiles of breath acetone